1 Feb 1999

umber Of Roots Of An Equation

Given an equation f(x) = 0, a number a such that f(a) = 0 is called a root of the equation.  In other words, a root is a solution of the equation.

To find the number of roots of an equation, usually graphical method is used.

  1. If y = f(x) is easy to sketch, the number of x-intercepts is the number of roots.
    2.
  2. Otherwise, rewrite the equation as g(x) = h(x) where the graphs of y = g(x) and y = h(x) are easy to sketch.   Then the number of intersection points between the two graphs is the number of roots of the equation.
    3.

xistence Of Roots In An Interval

If 
  • the graph of y = f(x) is continuous in [a, b] (ie, there is no break in the graph from a to b), and
  • f(a) and f(b) are of opposite sign,

then the equation f(x) = 0 has a root in the interval (a, b).

inear Interpolation

Theory:  Suppose we know that there is a root of the equation f(x) = 0 in the interval (a, b), where b - a is small.  Then in the ideal situation, the chord joining the points P(a, f(a)) and Q(b, f(b)) will be close to the curve y = f(x).  In this case, the x-intercept, c, of the chord PQ will be close to the root of the equation.

Formula:
af(b) - bf(a)
c =  ¾¾¾¾¾
f(b) - f(a)

ewton-Raphson Method

Theory:  Suppose we know that a root of f(x) = 0 is close to x = x1.  Then in the ideal situation, the tangent to the curve at x = x1 will be close to y = f(x) in the surrounding of x = x1.
Therefore, we can use the x-intercept, x2, of the tangent to be an approximation to the root of f(x) = 0.

Formula:
f(x1)
x2 = x1 - ¾¾¾
f ¢(x1)

nder-/Over- Estimation

Whether linear interpolation or Newton-Raphson method will give an under- or over- estimation depends greatly on the shape of the curve near the root.

By computing the signs (ie + or -) of  f ¢(x) and  f ²(x) on an interval, we can deduce the general shape of the curve y = f(x) on the interval.  The following table shows all the four cases.

  f ¢(x) > 0 f ¢(x) < 0
f ²(x) > 0
f ²(x) < 0

Just remember the following results and the table will become easy.

 

f ¢(x) > 0  Ž the curve is increasing
f ¢(x) < 0  Ž the curve is decreasing
 
f ²(x) > 0  Ž the curve concaves upwards
f ²(x) < 0  Ž the curve concaves downwards
 

Let's look at the case when f ¢(x) > 0 and f ²(x) > 0.

It is clear from the above diagram that linear interpolation (red chord) produces an under-estimation whereas the Newton-Raphson method (blue tangent) produces an overestimation.

Similarly, you may deduce the results for the other 3 cases.